Leyton Orient took an early lead after only 3 minutes with a powerful shot by Jason Demetriou to open the scoring.
Swindon should have gone on to win the game, but couldn't score despite having plenty of chances, and hit both the post and the bar.
The defeat means Swindon will have to wait until next weekend if they beat Carlisle, and other results go their way to see if they manage to avoid relegation for another season.